Polynomial Power Method: An Extension of the Standard Power Method to Para-Hermitian Matrices

Abstract

This paper expands the concept of the power method to polynomial para-Hermitian matrices in order to extract the principal analytic eigenpair. The proposed technique involves repeatedly multiplying the para-Hermitian matrix by a polynomial vector, followed by an appropriate normalization of the resulting product in each iteration, under the assumption that the principal analytic eigenvalue spectrally majorises the remaining eigenvalues. To restrain the growth in polynomial order of the product vector, truncation is performed after normalization in each iteration. The effectiveness of this proposed method has been verified through simulation results on an ensemble of randomly generated para-Hermitian matrices, demonstrating superior performance compared to existing algorithms.